On the Properties of Special Functions on the linear-type lattices

TL;DR

This paper develops a general theory for difference analogues of hypergeometric special functions on linear-type lattices, deriving recurrence relations and applying results to q-classical polynomials.

Contribution

It introduces a unified framework for hypergeometric-type functions on linear lattices and derives new difference-recurrence relations, with applications to q-polynomials.

Findings

Derived several difference-recurrence relations for special functions on linear lattices.

Provided integral representations of solutions to second order difference equations.

Applied the theory to q-classical polynomials.

Abstract

We present a general theory for studying the difference analogues of special functions of hypergeometric type on the linear-type lattices, i.e., the solutions of the second order linear difference equation of hypergeometric type on a special kind of lattices: the linear type lattices. In particular, using the integral representation of the solutions we obtain several difference-recurrence relations for such functions. Finally, applications to $q$-classical polynomials are given.